# 6.003 Homework Problems ```6.003 Homework #14
This homework will not be collected or graded. It is intended to help you practice for the
ﬁnal exam. Solutions will be posted.
Problems
1. Neural signals
The following ﬁgure illustrates the measurement of an action potential, which is an
electrical pulse that travels along a neuron. Assume that this pulse travels in the positive
z direction with constant speed ν = 10 m/s (which is a reasonable assumption for the
large unmyelinated ﬁbers found in the squid, where such potentials were ﬁrst studied).
Let Vm (z, t) represent the potential that is measured at position z and time t, where
time is measured in milliseconds and distance is measured in millimeters. The right
panel illustrates f (t) = Vm (30, t) which is the potential measured as a function of time t
at position z = 30 mm.
Axon
0
10
20
30
z [mm]
40
50
f(t) = Vm(30,t)
f(t) = Vm(30,t)
+50
0
-50
-100
0
2
4
6
Time [ms]
8
10
Part a. Sketch the dependence of Vm on t at position z = 40 mm (i.e., Vm (40, t)).
Part b. Sketch the dependence of Vm on z at time t = 0 ms (i.e., Vm (z, 0)).
Part c. Determine an expression for Vm (z, t) in terms of f (&middot;) and ν. Explain the relations
between this expression and your results from parts a and b.
Vm (z, t) =
6.003 Homework #14 / Fall 2011
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2. Characterizing block diagrams
Consider the system deﬁned by the following block diagram:
X
+
−
+
A
Y
−
3
2
1
2
a. Determine the system functional H =
A
A
Y
X.
b. Determine the poles of the system.
c. Determine the impulse response of the system.
6.003 Homework #14 / Fall 2011
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3. Bode Plots
Our goal is to design a stable CT LTI system H by cascading two causal CT LTI systems:
H1 and H2 . The magnitudes of H (jω) and H1 (jω) are speciﬁed by the following straightline approximations. We are free to choose other aspects of the systems.
|H(jω)| [dB]
−20
−40
ω [log scale]
1
10
100
1000
1
10
100
1000
|H1 (jω)| [dB]
24
6
ω [log scale]
a. Determine all system functions H1 (s) that are consistent with these design speciﬁ&shy;
cations, and plot the straight-line approximation to the phase angle of each (as a
function of ω).
b. Determine all system functions H2 (s) that are consistent with these design speciﬁ&shy;
cations, and plot the straight-line approximation to the phase angle of each (as a
function of ω).
6.003 Homework #14 / Fall 2011
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4. Controlling Systems
Use a proportional controller (gain K) to control a plant whose input and output are
related by
R2
F =
1 + R − 2R2
as shown below.
X
+
−
K
F
Y
a. Determine the range of K for which the unit-sample response of the closed-loop system
converges to zero.
b. Determine the range of K for which the closed-loop poles are real-valued numbers
with magnitudes less than 1.
6.003 Homework #14 / Fall 2011
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5. CT responses
We are given that the impulse response of a CT LTI system is of the form
A
t
T
where A and T are unknown. When the system is subjected to the input
x1 (t)
3
1
t
−2
the output y1 (t) is zero at t = 5. When the input is
x2 (t) = sin
πt
3
u(t),
the output y2 (t) is equal to 9 at t = 9. Determine A and T . Also determine y2 (t) for all
t.
6.003 Homework #14 / Fall 2011
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6. DT approximation of a CT system
Let HC1 represent a causal CT system that is described by
ẏC (t) + 3yC (t) = xC (t)
where xC (t) represents the input signal and yC (t) represents the output signal.
xC (t)
HC1
yC (t)
a. Determine the pole(s) of HC1 .
Your task is to design a causal DT system HD1 to approximate the behavior of HC1 .
xD [n]
HD1
yD [n]
Let xD [n] = xC (nT ) and yD [n] = yC (nT ) where T is a constant that represents the time
between samples. Then approximate the derivative as
dyC (t)
yC (t + T ) − yC (t)
=
.
dt
T
b. Determine an expression for the pole(s) of HD1 .
c. Determine the range of values of T for which HD1 is stable.
Now consider a second-order causal CT system HC2 , which is described by
ÿC (t) + 100yC (t) = xC (t) .
d. Determine the pole(s) of HC2 .
Design a causal DT system HD2 to approximate the behavior of HC2 . Approximate
derivatives as before:
dyC (t)
yC (t + T ) − yC (t)
y˙C (t) =
=
and
dt
T
y˙C (t + T ) − y˙C (t)
d2 yC (t)
=
.
2
dt
T
e. Determine an expression for the pole(s) of HD2 .
f. Determine the range of values of T for which HD2 stable.
6.003 Homework #14 / Fall 2011
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7. Feedback
Consider the system deﬁned by the following block diagram.
X
+
−
α
+
−
a. Determine the system functional
+
Y
po
Delay
Y
X.
b. Determine the number of closed-loop poles.
c. Determine the range of gains (α) for which the closed-loop system is stable.
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8. Finding a system
a. Determine the diﬀerence equation and block diagram representations for a system
whose output is 10, 1, 1, 1, 1, . . . when the input is 1, 1, 1, 1, 1, . . ..
b. Determine the diﬀerence equation and block diagram representations for a system
whose output is 1, 1, 1, 1, 1, . . . when the input is 10, 1, 1, 1, 1, . . ..
c. Compare the diﬀerence equations in parts a and b. Compare the block diagrams in
parts a and b.
6.003 Homework #14 / Fall 2011
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9. Lots of poles
All of the poles of a system fall on the unit circle, as shown in the following plot, where
the ‘2’ and ‘3’ means that the adjacent pole, marked with parentheses, is a repeated pole
of order 2 or 3 respectively.
Im z
&times;
(&times;)2
(&times;)3
Re z
&times;
Which of the following choices represents the order of growth of this system’s unit-sample
response for large n? Give the letter of your choice plus the information requested.
a. y[n] is periodic. If you choose this option, determine the period.
b. y[n] ∼ Ank (where A is a constant). If you choose this option, determine k.
c. y[n] ∼ Az n (where A is a constant). If you choose this option, determine z.
d. None of the above. If you choose this option, determine a closed-form asymptotic
expression for y[n].
6.003 Homework #14 / Fall 2011
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10.Relation between time and frequency responses
The impulse response of an LTI system is shown below.
h(t)
1
5
t
−1
If the input to the system is an eternal cosine, i.e., x(t) = cos(ωt), then the output will
have the form
y(t) = C cos(ωt + φ)
a. Determine ωm , the frequency ω for which the constant C is greatest. What is the
value of C when ω = ωm ?
b. Determine ωp , the frequency ω for which the phase angle φ is − π4 . What is the value
of C when ω = ωp ?
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6.003 Signals and Systems
Fall 2011